Chapter 6 - Set Theory - Exercise Set 6.3 - Page 372: 9

$For\,\,all\,\,sets\,\,A,\,B,\,\,and\,C\,\,\\if\,A \subseteq C\,and\,\,B \subseteq C\,then A \cup B \subseteq C.\\ this\,\,is\,\,true:\\ proof:\\ x\in A\cup B \Rightarrow x\in A\,or\,x\in B \\ (by\,def\,of\,union)\\ \because A \subseteq C\,and\, B \subseteq C\\ \therefore x\in A\,or\,x\in B\Rightarrow x\in C \\ \because x\in A\cup B\Rightarrow x\in C \\ \therefore A \cup B \subseteq C\\ $

$For\,\,all\,\,sets\,\,A,\,B,\,\,and\,C\,\,\\if\,A \subseteq C\,and\,\,B \subseteq C\,then A \cup B \subseteq C.\\ this\,\,is\,\,true:\\ proof:\\ x\in A\cup B \Rightarrow x\in A\,or\,x\in B \\ (by\,def\,of\,union)\\ \because A \subseteq C\,and\, B \subseteq C\\ \therefore x\in A\,or\,x\in B\Rightarrow x\in C \\ \because x\in A\cup B\Rightarrow x\in C \\ \therefore A \cup B \subseteq C\\ $